Abstract

This paper defines a matrix from which coherence property of imaging by partially coherent Koehler illumination is determined. The matrix termed coherency matrix in imaging system is derived by the space average of a product of a column vector and its transpose conjugate where each row of the column vector represents mutually incoherent light. The coherency matrix in imaging system has similar properties to the polarization matrix that is utilized for calculating the light intensity and degree of polarization of polarized light. The coherency matrix in imaging system enables us to calculate not only image intensity but also degree of coherence for image. Simulation results of the degree of coherence for image given by the coherency matrix in imaging system correspond to the complex degree of coherence obtained by the van Cittert-Zernike theorem.

Figures (11)

6f-imaging system assumed in this paper. In the figure, Lc, L′, and L′′ are lenses whose focal lengths are fc, f′, and f′′, respectively. The lens Lc is a condenser lens for Koehler illumination. The thick arrows with FT means the Fourier transform. Vector notations x=(x,y) and f=(f,g) are used.

(a) Simulation result of the modified H-S distance p by the illumination in Fig. 2 and a pair of ideal pinholes located at (0, 6.8736NA/λ) and (d, 0). The gray lines show positions where |j12(d)|=0 or 1; (b) Relationship between p and |j12(d)| when the diffraction effect is negligible.

Polarized light just after the pupil when the normal incident coherent illumination is polarized in f-direction and the object is an ideal pinhole. The NA of the imaging optics is set to 0.95 to emphasize the induced polarized electric field. (a) x-polarized electric field a^x(f,g); (b) y-polarized electric field a^y(f,g); (c) z-polarized electric field a^z(f,g).